A250122 Coordination sequence for planar net 3.12.12.

1, 3, 4, 6, 8, 12, 14, 15, 18, 21, 22, 24, 28, 30, 30, 33, 38, 39, 38, 42, 48, 48, 46, 51, 58, 57, 54, 60, 68, 66, 62, 69, 78, 75, 70, 78, 88, 84, 78, 87, 98, 93, 86, 96, 108, 102, 94, 105, 118, 111, 102, 114, 128, 120, 110, 123, 138, 129

Offset: 0

Author

Darrah Chavey, Nov 23 2014

Keywords

• nonn
• easy

Comments

Also, growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^6 = 1 >. See Magma program in A298805. - N. J. A. Sloane, Feb 06 2018

Links

• Maurizio Paolini, Table of n, a(n) for n = 0..1021
• Agnes Azzolino, Regular and Semi-Regular Tessellation Paper, 2011.
• Agnes Azzolino, Illustration of 3.12.12 tiling [From previous link]
• Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
• Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also on arXiv, arXiv:1803.08530 [math.CO], 2018-2019. See section 10 The 3.12^2 tiling.
• Rostislav Grigorchuk and Cosmas Kravaris, On the growth of the wallpaper groups, arXiv:2012.13661 [math.GR], 2020. See section 4.6 p. 23.
• Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
• Tom Karzes, Tiling Coordination Sequences
• Maurizio Paolini, C program for A250122
• Reticular Chemistry Structure Resource, hca
• N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
• Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

Crossrefs

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).

Cf. A298805.

Programs

Mathematica

Join[{1, 3, 4}, LinearRecurrence[{2, -3, 4, -3, 2, -1}, {6, 8, 12, 14, 15, 18}, 100]] (* Jean-François Alcover, Aug 05 2018 *)

Formula

From Joseph Myers, Nov 28 2014: (Start)

Empirically,

a(4n) = 10n - 2 except for a(0) = 1

a(4n+1) = 9n + 3

a(4n+2) = 8n + 6 except for a(2) = 4

a(4n+3) = 9n + 6. (End)

If these are correct, the sequence has g.f.

-(-1 - x - x^2 - 3*x^3 + x^4 - 5*x^5 + 3*x^6 - 4*x^7 + 2*x^8)/((x - 1)^2*(x^2 + 1)^2). - N. J. A. Sloane, Nov 28 2014

All the above conjectures are true. - N. J. A. Sloane, Dec 31 2015

E.g.f.: (9*x*cosh(x) - 4*(2*cos(x) + x^2 - 3) + 9*x*sinh(x) - (x - 3)*sin(x))/4. - Stefano Spezia, Jan 05 2023

Extensions

a(8) onwards from Maurizio Paolini and Joseph Myers (independently), Nov 28 2014